Abstract

The truncated second-order moments and generalized M2 factor (MG2 factor) of two-dimensional beams in the Cartesian coordinate system are extended to the case of three-dimensional rotationally symmetric hard-edged diffracted beams in the cylindrical coordinate system. It is shown that the propagation equations of truncated second-order moments and the MG2 factor take forms similar to those for the nontruncated case. The closed-form expression for the MG2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams is derived that depends on the truncation parameter β and beam order N. For N, the MG2 factor equals 4/3 corresponding to the value of truncated plane waves, which guarantees consistency of the formalism.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  2. International standards organization (ISO) Document, “Lasers and laser-related equipment-test methods for laser beam parameters-beam widths, divergence angle and beam propagation factor,” (ISO, Geneva, Switzerland, 1999).
  3. A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  4. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  5. D. Ding, X. Liu, “Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
    [CrossRef]
  6. A. Belafhal, M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
    [CrossRef]
  7. B. Lü, S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
    [CrossRef]
  8. R. Martı́nez-Herrero, P. M. Mejı́as, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  9. R. Martı́nez-Herrero, P. M. Mejı́as, M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef] [PubMed]
  10. R. Martı́nez-Herrero, P. M. Mejı́as, S. Bosch, A. Carnicer, “Spatial width and power-content ratio of hard-edge diffracted beams,” J. Opt. Soc. Am. A 20, 388–391 (2003).
    [CrossRef]
  11. C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
    [CrossRef]
  12. S. Amarande, A. Giesen, H. Hügel, “Propagation analysis of self-convergent beam width and characterization of hard-edge diffracted beams,” Appl. Opt. 39, 3914–3924 (2000).
    [CrossRef]
  13. B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
    [CrossRef]
  14. B. Lü, S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2101 (2001).
    [CrossRef]
  15. S. A. Collins, “Lens-system diffraction integral written terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  16. A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 51, 431; Vol. 1, p. 137.
  17. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  18. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]

2003 (1)

2001 (2)

B. Lü, S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2101 (2001).
[CrossRef]

B. Lü, S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
[CrossRef]

2000 (3)

A. Belafhal, M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
[CrossRef]

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

S. Amarande, A. Giesen, H. Hügel, “Propagation analysis of self-convergent beam width and characterization of hard-edge diffracted beams,” Appl. Opt. 39, 3914–3924 (2000).
[CrossRef]

1999 (1)

1996 (2)

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

1995 (1)

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1993 (1)

1988 (1)

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1970 (1)

Amarande, S.

Ambrosini, D.

Arias, M.

Bagini, V.

Belafhal, A.

A. Belafhal, M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
[CrossRef]

Belanger, P.-A.

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

Borghi, R.

Bosch, S.

Breazeale, M. A.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Carnicer, A.

Collins, S. A.

Ding, D.

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 51, 431; Vol. 1, p. 137.

Giesen, A.

Gori, F.

Hügel, H.

Ibnchaikh, M.

A. Belafhal, M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
[CrossRef]

Liu, X.

Lü, B.

B. Lü, S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
[CrossRef]

B. Lü, S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2101 (2001).
[CrossRef]

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

Luo, S.

B. Lü, S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
[CrossRef]

B. Lü, S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2101 (2001).
[CrossRef]

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

Marti´nez-Herrero, R.

Meji´as, P. M.

Pacileo, A. M.

Pare, C.

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

Santarsiero, M.

Schirripa Spagnolo, G.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Sigeman, A. E.

A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

Wen, J. J.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (1)

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Mod. Opt. (1)

B. Lü, S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2169–2178 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (4)

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh–Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

A. Belafhal, M. Ibnchaikh, “Propagation properties of Hermite–Cosh–Gaussian laser beams,” Opt. Commun. 186, 269–276 (2000).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Opt. Lett. (2)

Other (4)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

International standards organization (ISO) Document, “Lasers and laser-related equipment-test methods for laser beam parameters-beam widths, divergence angle and beam propagation factor,” (ISO, Geneva, Switzerland, 1999).

A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

A. Erdelyi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, pp. 51, 431; Vol. 1, p. 137.

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Figures (1)

Fig. 1
Fig. 1

Generalized MG2 factor as a function of truncation parameter β and beam order N.

Equations (57)

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r2=2πI00ar2|E(r, 0)|2rdr,
p2=2πk2I00a|E(r, 0)|2rdr+32π3k2I0|E(a, 0)|2,
rp=πikI00a{r[E(r, 0)]*E(r, 0)-rE(r, 0)E*(r, 0)}rdr,
I0=2π0a|E(r, 0)|2rdr
E(r2, z)=ikBexp-ikDr222B0aE(r, 0)×exp-ikAr22BJ0kr2rBrdr
(B0),
I2(r2, z)=|E(r2, z)|2=2π0a1h(r1)J0(kr1r2)r1dr12,
a1=a/B,
r1=r/B,
h(r1)=iBλexp-ikABr122E(Br1, 0).
r22=2πk2I0a1|h(r1)|2r1dr1+32π3k2I|h(a1)|2,
r2p2=πikI0a1{r1h*(r1)h(r1)-r1[h(r1)]*h(r1)}r1dr1+Dr22B,
I=2π0a1|h(r1)|2r1dr1.
I=I0/λ2.
r22=A2r2+B2p2+2ABrp,
r2p2=ACr2+BDp2+(AD+BC)rp.
0cos(ax2)J0(bx)J0(xy)xdx=12asinb2+y24aJ0by2a,a>0,b>0,
0sin(ax2)J0(bx)J0(xy)xdx=12acosb2+y24aJ0by2a,a>0,b>0,
E˜(p2, z)=2π expikBp222D1D0aE(r, 0)×exp-ikCr22DJ0krp2Drdr.
I˜2(p2, z)=|E˜(p2, z)|2=2π0a˜1h˜(r˜1)J0(kr˜1p2)r˜1dr˜12,
r˜1=r/D,
a˜1=a/D,
h˜(r˜1)=D exp-ikCDr˜122E(Dr˜1, 0).
p22=2πk2I˜0a˜1|h˜(r˜1)|2r˜1dr˜1+32π3k2I˜|h(a˜1)|2,
I˜=2π0a˜1|h˜(r˜1)|2r˜1dr˜1.
I˜=I0.
p22=C2r2+D2p2+2CDrp.
MG2=k(r22p22-r2p22)1/2=k(r2p2-rp2)1/2,
E(r, 0)=exp-(N+1)r2w02n=0N1n!(N+1)r2w02n,
γ(α, x)=0xexp(-t)tα-1dt,
r2=w022(N+1)C2C1,
p2=4(N+1)k2w02C3C1,
rp=0,
C1=n1=0Nn2=0N2-(n1+n2)n1!n2!γ[n1+n2+1, 2(N+1)β2],
C2=n1=0Nn2=0N2-(n1+n2)n1!n2!γ[n1+n2+2, 2(N+1)β2],
C3=2-(2N+1)(N!)2γ[2N+2, 2(N+1)β2]+163exp[-2(N+1)β2]×n=0N1n![(N+1)β2]n2,
β=aw0(truncationparameter).
MG2=2C2C3C1.
M2=MG2(β)=(2N+1)!n1=0Nn2=0N(n1+n2+1)!2-(n1+n2)n1!n2!1/22NN!n1=0Nn2=0N(n1+n2)!2-(n1+n2)n1!n2!.
MG2={[1+2β2-exp(2β2)][2β2-29/3-exp(2β2)]}1/2exp(2β2)-1.
M2=MG2(β)=1,
E(r, 0)=1,ra0,r>a.
r2=a2/2,
p2=32/3k2a2,
rp=0.
MG2=4/3.
r2=0r2|E(r, 0)|2rdr0|E(r, 0)|2rdr,
p2=1k20|E(r, 0)|2rdr0|E(r, 0)|2rdr,
rp=12ik0a{r[E(r, 0)]*E(r, 0)-rE(r, 0)E*(r, 0)}rdr0|E(r, 0)|2rdr.
0exp(-pt)tvdt=Γ(v+1)p-v-1,
r2=w022(N+1)×n1=0Nn2=0N2-(n1+n2)n1!n2!Γ(n1+n2+2)n1=0Nn2=0N2-(n1+n2)n1!n2!Γ(n1+n2+1),
p2=2(N+1)k2w02×2-2NΓ(2N+2)(N!)2n1=0Nn2=0N2-(n1+n2)n1!n2!Γ(n1+n2+1),
rp=0.
Γ(n+1)=n!
r2=w022(N+1)n1=0Nn2=0N(n1+n2+1)!2-(n1+n2)n1!n2!n1=0Nn2=0N(n1+n2)!2-(n1+n2)n1!n2!,
p2=2(N+1)k2w02(2N+1)!2-2N(N!)2n1=0Nn2=0N(n1+n2)!2-(n1+n2)n1!n2!,
M2=(2N+1)!n1=0Nn2=0N(n1+n2+1)!2-(n1+n2)n1!n2!1/22NN!n1=0Nn2=0N(n1+n2)!2-(n1+n2)n1!n2!.

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